This is a complex topic with strong emotions occasionally fuelling the debate due to the role Christianity plays in the modern world, but ill give it a go. The first part to your question is why the Christians were persecuted. The Roman world was one of many religions, even within the typical pantheon of Jupiter and Ares there were many variations which people often had personal or familial - or even geographical (particular towns supporting certain cults) - attachments to. Behind this generic group gods, however, there were also other religions in the Empire, Judaism being one of them. Judaism was seen as an ‘older’ religion, and respected by Roman authorities (to an extent), but Christianity was new and, most importantly, was seen as ‘different’ and ‘other’. Christians did not commit blood sacrifice, as other major religions had done, nor did they worship outside their temples (churches), and so they were isolated from the rest of the religious community (which was literally everyone). This created the same situation that is experienced in the modern world within some societies as a fear of difference, which becomes a hatred. In one way or another, this hatred turned to persecution.

Flashing forward to the end of the crisis of the third century, Diocletian became emperor, and for 20 years tried to rejuvenate the Roman Empire. He reformed military, economic, and political aspects of the Roman world, and then for some reason began one of the worst periods of persecution in Roman history. Scholarship has tried to show his motivations for such actions, which has either pointed to the personal ‘pagan’ piety of Diocletian, or that of his junior emperor, Galerius. In this view, persecution was just the final part of a long series of reforms to reorder the Roman world. However, it may also have been due to the influence of pagan intellectuals like Porphyry, who were eager to show the weaknesses of early Christianity and spread hatred amongst the general population towards Christians. In this view, Persecution was undertaken as a reaction to negative public sentiment. Now widespread persecution must have been expensive; there were the actual laws/edicts which needed to be written (by lawyers employed by emperors), then it had to be promulgated across the Empire, and finally (and probably most expensively), it had to be enforced. So, it might have benefitted Diocletian/Galerius more to commit to a token persecution where Empire-side edicts are passed but not truly enforced. This was the case for Diocletian’s Great Persecution (AD 303-305), during which recent scholarship has suggested fewer than 5% of the Christian population was killed - even then it was mostly deaths of voluntary martyrs (Rees, 2004). This is exemplified by the situation in Caesarea, a city which had a massive Christian population, but where only nine Christians were actually martyred - and four of them voluntarily at that (Eusebius, Ecclesiastical History).

To answer your question then, one reason why Persecution might have failed to eradicate Christianity was because the emperors simply never aimed to destroy it, but just wanted to appeal to anti-Christian public views to stay popular. However, another view (and another reason why Diocletian’s Great Persecution might have failed) is that it was simply beyond the empire’s resources to eradicate such a fast-growing and widespread religion. This is the most popular view, and the one argued by Keresztes (1983) and Davies (1989). But you can make up your own mind based on the example I have laid out, or through reading the literature that I have laid out below.

Bibliography:

Eusebius, Ecclesiastical History, tr. H.J. Lawlor (London, 1927).

Davies, P.S. (1989), ‘The Origin and Purpose of the Persecution of AD 303’, Journal of Theological Studies 40.1.

Hopkins, K. (2000), ‘A World Full of Gods’ (Cambridge).

Keresztes, P. (1983), ‘From the Great Persecution to the Peace of Galerius’, Vigiliae Christianae 37.4.

Rees, R. (2004), ‘Diocletian and the Tetrarchy’ (Edinburgh).